Dimension-six matrix elements for meson mixing and lifetimes from sum rules

The hadronic matrix elements of dimension-six $\Delta F=0,2$ operators are crucial inputs for the theory predictions of mixing observables and lifetime ratios in the $B$ and $D$ system. We determine them using HQET sum rules for three-point correlators. The results of the required three-loop computation of the correlators and the one-loop computation of the QCD-HQET matching are given in analytic form. For mixing matrix elements we find very good agreement with recent lattice results and comparable theoretical uncertainties. For lifetime matrix elements we present the first ever determination in the $D$ meson sector and the first determination of $\Delta B=0$ matrix elements with uncertainties under control - superseeding preliminary lattice studies stemming from 2001 and earlier. With our state-of-the-art determination of the bag parameters we predict: $\tau(B^+)/\tau(B_d^0) = 1.082_{-0.026}^{+0.022}$, $\tau(B_s^0)/\tau(B_d^0) = 0.9994\pm0.0025$, $\tau(D^+)/\tau(D^0) = 2.7_{-0.8}^{+0.7}$ and the mixing-observables in the $B_s$ and $B_d$ system, in good agreement with the most recent experimental averages.


Introduction
The mixing of neutral mesons proceeds through flavour-changing neutral currents and is therefore loop suppressed in the Standard Model. Thus, mixing observables are very sensitive to new physics effects. Our ability to constrain new contributions strongly relies on a high degree of precision in both experiment and theory. Mixing is most pronounced in the B s system where the relative decay rate difference amounts to about 13%. Here the experimental precision has surpassed the theoretical one by a significant margin [1].
The theory expression for mixing observables is a product of perturbative coefficients and non-perturbative matrix elements. The perturbative part is known up to NLO-QCD (see the discussion below) and first steps in the direction of a NNLO-QCD evaluation have recently been performed by [2]. However, the dominant theoretical uncertainties still stem from hadronic matrix elements of local ∆B = 2 four-quark operators. They are usually determined by lattice simulations and results for the leading dimension-six operators are available from several collaborations [3][4][5]. If only the latest lattice results [5] are used, small tensions at the level of two sigma emerge in B s mixing [5,6]. To either settle or solidify this issue, an independent determination of the matrix elements and further scrutinization of the theoretical methods are necessary. We address both these points in this paper.
An alternative way to determine hadronic matrix elements is given by QCD sum rules [7,8]. This approach employs quark-hadron duality and the analyticity of Green functions instead of the discretization of space-time. Thus, its sources of uncertainties are entirely different from lattice simulations and sum rule analyses can provide truly independent results. We determine the hadronic matrix elements of the dimension-six ∆B = 2 operators for B-mixing from a sum rule for three-point correlators first introduced in [9]. The sum rule is valid at scales µ ρ ∼ 1.5 GeV which are much smaller than the bottom-quark mass. Therefore the sum rule is formulated in HQET, where quantum fluctuations with a characteristic scale of the order of the bottom-quark mass have been integrated out. We then run the HQET matrix elements up to a scale µ m of the order of the bottom-quark mass where the matching to QCD can be performed without introducing large logarithms. Earlier sum rule results are available for the SM operator Q 1 [10,11] and condensate corrections have been computed for dimension-six [11][12][13][14][15] and seven [12,13] operators. The same strategy is then applied to determine the matrix elements of dimension-six ∆B = 0 operators, which are the non-perturbative input for calculating ratios of lifetimes of different mesons, like τ (B + )/τ (B d ) and τ (B s )/τ (B d ), see e.g. [16] for a review. Here the perturbative part of the prediction is also known to NLO-QCD.
The theory prediction for the B s decay rate difference ∆Γ s and for ratios of lifetimes of different B mesons is based on the Heavy Quark Expansion (HQE) [17][18][19][20]. The HQE is an OPE in the Minkowski domain which has fuelled speculations about large violations of duality, in particular for ∆Γ s which is dominated by the b → ccs transition. 1 A recent confrontation of HQE predictions with experiment has ruled out duality violations larger than about 20% [6]. Ratios of meson lifetimes are a good testing ground for the validity of the HQE, but have suffered from large hadronic uncertainties [16] in the past because only outdated lattice results [22,23] for the required ∆B = 0 matrix elements of four quark operators were available. We present the first state-of-the-art calculation of the ∆B = 0 matrix elements and determine the lifetimes with significantly reduced uncertainties.
In the charm sector the validity of the HQE is rather uncertain due to its smaller mass m c ∼ m b /3. The direct translation of the predictions for B mixing fails by several orders of magnitude [24]. However it has been argued that higher-dimensional contributions can lift the severe GIM suppression in the charm sector and potentially explain the size of mixing observables [24][25][26][27][28]. D-meson lifetimes have been studied recently [29] and have shown no indications for a breakdown of the HQE, albeit with large hadronic uncertainties. We translate our sum rule results to the charm sector as well. The ∆C = 2 matrix elements show good agreement with lattice results [30][31][32] and the ∆C = 0 results are used to update the D + − D 0 lifetime ratio.
The outline of this work is as follows: In Section 2 we describe the details of the QCD-HQET matching computation focussing on ∆B = 2 operators. The sum rule and the calculation of the three-point correlators are discussed in Section 3. Our results for the matrix elements are presented in Section 4 and compared to other recent works. In Section 5 we study ∆B = 0 operators and ratios of B-meson lifetimes. We determine the matrix elements of ∆C = 0, 2 operators in Section 6 and update the HQE result for the D + − D 0 lifetime ratio using these results. Finally, we conclude in Section 7.

QCD-HQET matching for ∆B = operators
We perform the matching computation between QCD and HQET operators at the one-loop level. The details of the computation are described in Section 2.1 for the ∆B = 2 operators. Our results for the matching of the operators and Bag parameters are given in Section 2.2 and Section 2.3, respectively.

Setup
The matching calculation for the SM operator Q 1 appearing in ∆M s has been performed in [33][34][35]. We compute the matching coefficients of the full dimension-six ∆B = 2 operator basis needed for ∆M s in BSM theories and for ∆Γ s in the SM. We work in dimensional regularization with d = 4 − 2 and an anticommuting γ 5 (NDR scheme). We consider the following operators in QCD To fix the renormalization scheme we also have to specify a basis of evanescent operators [36][37][38]. We do this following [39]. The explicit form of the evanescent operators can be found in Appendix A. On the HQET side, we have the operators where the HQET field h (+) (x) annihilates a bottom quark, h (−) (x) creates an antibottom and we have introduced the notation Note that no operatorQ 3 appears on the HQET side because it is not linearly independent, just like its QCD equivalent at leading order in 1/m b [40]. We define the evanescent HQET operators up to three constants a i with i = 1, 2, 3 which allow us to keep track of the scheme dependence. Again the explicit basis of the evanescent operators can be found in Appendix A. The matching condition for the ∆B = 2 operators is given by where A = B |A|B . The matching coefficients can be expanded in perturbation theory and take the form Thus the matching calculation can be performed with external quark states. The partonic QCD matrix elements are where we sum over O, including all physical and evanescent operators, and the color singlet initial and final state have been projected out. The two tree-level contractions appear with a relative minus sign. The gluon corrections are shown in Figure 1 and do not contain self-energy insertions on the external legs, since the quark fields are renormalized in the on-shell scheme. The HQET matrix elements follow from the replacements Q →Q, O →Õ, Z OS b → Z OS h and using HQET propagators instead of the full QCD ones for the bottom quark. The heavy quark on-shell renormalization constants are The light-quark renormalization is trivial in the massless case Z OS q = 1. For the renormalization of the physical operators the MS scheme is used. In accordance with [36][37][38] the evanescent operators are renormalized by a finite amount such that their physical matrix elements vanish. Consequently the Wilson coefficients C Q iẼj are not required for the determination of the hadronic matrix elements and are omitted in the results shown below. However, in the matching computation itself the matrix elements are taken between external on-shell quark states and are therefore not IR finite. While the IR divergences cancel in the matching of the QCD to the HQET operators there are non-vanishing contributions to the physical matching coefficients C QQ from matrix elements of the evanescent operators that are multiplied by IR poles since the evanescent operators are defined differently in QCD and HQET, cf. Appendix A.
We also find that the NLO matching coefficients C of the operator Q 3 are affected by the finite renormalization of the evanescent operatorẼ 2 which contains contributions proportional to the physical operators. This usually only happens at NNLO (as is the case for the other operators) but is already present here at NLO because the tree-level matching coefficient C of this operator is non-vanishing and, therefore, the NLO matrix element of the evanescent HQET operatorẼ 2 already appears at NLO in the matching calculation.
In the computation we have used both a manual approach and an automated setup utilizing QGRAF [41] and Mathematica to generate the amplitudes. The Dirac algebra has been performed with a customized version of TRACER [42] as well as with Package-X [43,44] and the QCD loop integrals have been evaluated using Package-X [43,44]. We have also checked our results by performing the calculation with a gluon mass as an IR regulator and found full agreement.

Results
We write the LO QCD anomalous dimension matrix (ADM) as where γ QQ is the ADM for the physical set of operators (1), γ QE describes the mixing of the physical operators into the evanescent ones (80), γ (0) EQ vanishes (see [38]) and γ (0) EE is not required. We decompose the LO HQET ADMγ (0) analogously. Our results for the non-vanishing entries are given in Appendix A.
The non-vanishing Wilson coefficients at LO are The NLO corrections to the matching coefficients read where L µ = ln(µ 2 /m 2 b ) and we have set N c = 3 to keep the results compact.

Matching of QCD and HQET Bag parameters
We define the QCD bag parameters B Q following [45] where the coefficients read the B meson decay constant f B is defined as M B is the mass of the B meson and B Q i = 1 corresponds to the VSA approximation. We note that the quark masses appearing in (12) are not MS masses which is the usual convention today [5,46], but pole masses. We prefer the definition (11) for the analysis because the use of MS masses makes the LO ADM of the Bag parameters explicitly µ-dependent and prohibits an analytic solution of the RGE. At the end we convert our results to the convention of [5,46] which we denote as where the A Q (µ) follow from A Q with the replacements m b → m b (µ) and m q → m q (µ). Similar to (11), we use for the HQET operators where and the matrix elements have been taken between non-relativistically normalized such that The parameter F (µ) is defined as and related to the decay constant by with [47] From (11) and (15), we obtain, using (4), (17) and (20), The HQET bag parameters BQ are determined from a sum rule analysis.

HQET sum rule
The HQET sum rule is introduced in Section 3.1. We give results for the doublediscontinuity of the three-point correlators in Section 3.2 and describe the determination of HQET and QCD Bag parameters in Section 3.3.

The sum rule
We define the three-point correlator where ω 1,2 = p 1,2 · v andj are interpolating currents for the pseudoscalar B and B mesons. The correlator (23) is analytic in ω 1,2 apart from discontinuities for positive real ω. This allows us to construct a dispersion relation where ρQ is the double discontinuity of KQ in ω 1 and ω 2 . The second term on the right originates from the integration of KQ along the circle at infinity in the complex η 1 or (and) η 2 planes and is therefore polynomial in ω 1 or (and) ω 2 . The correlator KQ can be computed by means of an OPE for values of ω 1,2 that lie far away from the physical cut. Assuming quark-hadron duality, we can equate the correlator K OPẼ Q with its hadronic counterpart which is obtained from integration over the hadronic spectral function We use a double Borel transformation with respect to ω 1,2 to remove the contribution from the integration over the circle at infinity and to suppress the sensitivity to the continuum part ρ cont Q of the spectral function, which yields the sum rule Figure 2: Leading order diagram for the three-point HQET correlator (23). The sum over the two possible contractions of the operatorQ is implied.
In principle one can proceed by modelling the continuum ρ cont Q . The desired matrix element of the operatorQ between the mesonic ground state can then be disentangled by varying the Borel parameters. However, the continuum contribution is exponentially suppressed in the Borel sum rule and it is safe to simply "cut off" the sum rule by assuming that which directly yields a finite-energy sum rule for the matrix elements Thus, the determination of the HQET Bag parameters requires the computation of the spectral functions ρ OPẼ Q . The leading condensate corrections have been determined in [12,13]. We compute the O(α s ) corrections to the perturbative contribution below.

Spectral functions at NLO
We determine the spectral functions by first computing the correlator and then taking its double discontinuity. At LO we have to evaluate the diagram in Figure 2 which factorizes into two two-point functions. We obtain 2 Figure 3: Diagrams contributing to the three-point HQET correlator (23) at NLO. Symmetric diagrams are not shown. where is the LO result for the two-point correlator where ω = p · v and the use ofμ 2 = µ 2 exp(γ E )/(4π) corresponds to the MS scheme. The bare NLO correction K (1),barẽ Q is given by the diagrams shown in Figure 3. At this order we get corrections that do not factorize due to gluon exchange between the left and right-hand side. These genuine three-loop contributions -given by the diagrams in the second row of Figure 3 -are the most computationally challenging. The Dirac traces have been evaluated with both TRACER [42] and Package-X [43,44]. We use the code FIRE [48][49][50] to find IBP relations [51] between the three-loop integrals and to reduce them to a set of master integrals via the Laporta algorithm [52]. The relevant master integrals have been computed analytically in [11,53]. 2 As discussed below the sum rule reproduces the VSA at LO. Therefore the factors AQ i appear at leading order in the expansion of the results in . However, the correlator is computed in d dimensions and corrections can appear. We find that this happens only forQ 1 where the contraction of the two γ matrices inside the trace yields a d-dimensional factor.
The renormalized NLO correlators are given by whereγ (0) j = −3C F is the LO anomalous dimension of the currentsj ± . The contributions from the evanescent operators modify the double discontinuities of the correlators by a finite amount and introduce a dependence of the correlator on the choice of basis of the HQET evanescent operators. This dependence propagates to the HQET bag parameters extracted in the sum rule and cancels with the HQET evanescent scheme dependence of the matching coefficients (10) in the matching equation (22) for the QCD Bag parameters. The results for the bare correlators are available as an ancillary Mathematica file with the arXiv version of this article. Here, we only show the compact results for the double discontinuities of the correlators.
Methods to compute the double discontinuities of the correlators have been described in [11,54]. The results take the form where is the discontinuity of the two-point correlator (35) up to two-loop order [55][56][57].
The non-factorizable contributions are where x = ω 2 /ω 1 , L ω = ln(µ 2 /(4ω 1 ω 2 )) and we obtain where Taking a 2 = −4 in accordance with [11] we reproduce their result for rQ 1 up to a factor of 2 which is due to the different normalization of the HQET operators.

Sum rule for the Bag parameters
Inserting the decomposition (37) into the sum rule (31) allows us to subtract the factorized contribution using the sum rule [55][56][57] for the HQET decay constant The factorizable part of (37) exactly reproduces the VSA for the matrix elements.
After subtracting it, we obtain a sum rule for the deviation ∆BQ = BQ − 1 from the VSA. In the traditional sum rule approach this gives .
The stability of the sum rule (44) can then be assessed numerically by variation of the cutoff ω c and the Borel parameters t i , see e.g. [54,56]. In our analysis we follow a different approach that allows us to obtain analytic results for the HQET Bag parameters. This exploits the fact that the dispersion relation (25) is not violated by the introduction of an arbitrary weight function w(ω 1 , ω 2 ) in the integration as long as it is chosen such that no additional discontinuities appear in the complex plane. 3 In the presence of such a weight function w the square of the sum rule (42) takes the form Since the condensate contributions have already been taken into account in [11][12][13] and are in the subpercent range we only focus on the perturbative contribution to the sum rule. By using (45) with the choice we can remove the integration in (43) altogether and find the simple result The sum rule is valid at a low scale µ ρ ∼ 2ω i ∼ 2Λ where the logarithms that appear in the spectral functions are small. From there we have to evolve the results for the Bag parameters up to the scale µ m ∼ m b where the matching (22) to the QCD Bag parameters can be performed without introducing large logarithms. From (15) and the running of the HQET operators and decay constant we obtain the RG equations for the HQET Bag parameters whereÂQ is the diagonal matrix with entries AQ given in (16). The LO solution to (49) takes the form with the LO evolution matrix whereV is the transformation that diagonalizes the ADMγ and the vector γ . As part of our error analysis we allow the matching scale µ m to differ from m b (m b ) and then evolve the QCD Bag parameters back to m b (m b ). The LO evolution matrix has the same form as its HQET counterpart (51) while the anomalous dimension matrix of the QCD Bag parameters is given byγ We only resum the leading logarithms because the NLO anomalous dimensions in HQET are currently not known. This implies that dependence of the QCD matrix elements on the basis of evanescent HQET operators does not fully cancel. As discussed below, we use variation of the parameters a i to estimate the effects of NLL resummation. We expect this effect to be small since the scales µ ρ and µ m are not very widely separated and ln(µ m /µ ρ ) is of order one.

Results for ∆B = 2 operators
We describe our analysis in Section 4.1 and give the results for the Bag parameters, together with a comparison with other works, in Section 4.2. In Section 4.3 the results for the mixing observables with our Bag parameters are shown.

Details of the analysis
We determine the HQET Bag parameters from the sum rule (47) with the central values µ ρ = 1.5 GeV and Λ = 0.5 GeV. We use RunDec [58,59] to evolve α s (M Z ) = 0.1181 [60] down to the bottom-quark MS mass m b (m b ) = 4.203 GeV [61,62] with five-loop accuracy [63][64][65][66][67]. From there we use two-loop running with four and five flavours in HQET and QCD, respectively. The decoupling of the bottom quark is trivial at this accuracy. The HQET Bag parameters are then evolved from the scale µ ρ up to the scale µ m = m b (m b ) using (50). There the matching (22) to the QCD Bag parameters is performed. The factors C Q iQj (µ)/C 2 (µ) are expanded in α s and truncated after the linear term. We also expand the ratios AQ j /A Q i strictly in Λ/m b and m q /m b . Up to higher order perturbative corrections, this is equivalent to the use of the VSA for the power-suppressed HQET operators that arise in the QCD-HQET matching (4).
A small dependence on the choice of basis for the evanescent HQET operators remains in the QCD Bag parameters because the RG evolution of the HQET Bag parameters is only known at the LL level. We have checked that the a i -dependence fully cancels when the scales µ ρ and µ m are identified and the matching (22) is strictly expanded in the strong coupling, which serves as a strong cross-check of our calculation. For different scales µ ρ and µ m the remaining a i -dependence can be removed by a future computation of the NLO ADMs.
Finally, we convert the QCD Bag parameters B Q to the usual convention B Q defined in (14). This is done by expanding the ratios of the prefactors in α s and truncating them after the linear term.
To estimate the errors of the Bag parameters we take the following sources of uncertainties into account: • The uncertainty in the analytic form (47)  • The condensate contributions to BQ 1 and BQ 2 are taken from [12,13] and are in the subpercent range. For BQ 4 and BQ 5 , which have not been determined there, we therefore add an error of ±0.01 to the perturbative results.
• To assign an uncertainty from the unknown α 2 s contributions to the spectral densities we vary the scale µ ρ in the range [1,2] GeV.
• As discussed above we implicitly include higher-order corrections in 1/m b in the VSA approximation. The non-factorizable corrections of this kind are of the order (α s /π) · (Λ/m b ) ∼ 0.01, which we take as an estimate for the error.
• Higher order perturbative contributions to the QCD-HQET matching relation and the RG evolution of the Bag parameters are estimated through variation of µ m in the range [3,6] GeV and variation of the a i in the range [-10,10]. The QCD Bag parameters are then evolved to the central scale m b (m b ) with LL accuracy as described in Section 3.3.
The variation of µ m by the usual factors of 1/2 and 2 would lead to a doubling of the matching uncertainty estimates given below, which would significantly exceed the effect of the NLO matching at the central scale. We therefore use a less conservative range but cannot exclude larger matching effects at NNLO at present, while a calculation is not available.
• The parametric uncertainty from α s (M Z ) is in the permille range and neglected.
The individual errors are then summed in quadrature. We also divide the uncertainties into a sum rule uncertainty which contains the first three items in the list above and a matching uncertainty which contains the remaining three.

Results and comparison
From the sum rule we obtain the HQET Bag parameters where we have set a i = 0 for i = 1, 2, 3 to specify a unique basis of evanescent HQET operators. The individual uncertainties were determined as described above and added in quadrature. The corrections to the VSA for scales in the range from 1 -2 GeV are at the level of 5 -11 % forQ 1,2 and 0 -4 % forQ 4,5 . We find that the total sum rule uncertainties of the Bag parameters are quite small. This is because the sum rule (47) is formulated for the deviation from the VSA and the substantial relative uncertainties of the sum rule itself are small in comparison with the VSA contribution to the Bag parameters.
Following the steps outlined in Section 4.1 we obtain the following results for the QCD Bag parameters The evolution to the scale m b (m b )) and the matching to QCD increase the deviations from the VSA to up to 18 %. With the exception of B Q 3 the uncertainties of the Bag parameters are dominated by the matching. A detailed list of the uncertainties can be found in Appendix B.
parameters of the operators Q 1 , Q 2 and Q 3 . The uncertainties of our sum rule analysis are similar to those obtained on the lattice. We observe that the uncertainty of the Bag parameter B Q 3 is significantly larger than those of B Q 1 and B Q 2 . This is related to the small color factor A Q 3 = 1/3 + O(1/m b ) which implies that the sum rule uncertainties get enhanced by the factors AQ (22) of the Bag parameters. The absolute sum rule uncertainty of the matrix element of Q 3 is of a similar size as that of the other operators.
The tiny difference of the central value of B Q 1 compared to the sum rule determination [11] is mostly due to different scale choices. Since B Q 1 does not run at the LL order, [11] sets all scales equal to the bottom-quark mass. We, however, evaluate the sum rule at a lower scale µ ρ ∼ 1.5 GeV where the strong coupling is larger and causes a bigger deviation from the VSA.
Only two previous lattice results [4,5] exist for the matrix elements of the operators Q 4 and Q 5 , and they differ at the level of more than two sigma. Our results are in very good agreement with those of [5] and show an even higher level of tension with [4] in B Q 5 .

B s and B d mixing observables
where f is a flavor-specific final state, i.e.B s → f and B s →f are forbidden (see [1] for a recent review of B s mixing). Using our values for the Bag parameters, we give predictions for these observables and compare them to the current experimental averages given by the HFLAV [69]. In our sum rule determination we have assumed the light quark q in the B q meson to be massless. The corrections to (47) from a non-zero strange-quark mass are of the order (α s /π)(m s /(2Λ)) ≈ 0.02. This point has recently been discussed in more detail in [70]. We add another uncertainty of ±0.02 in quadrature to the results (55) to account for the unknown corrections. The effect on the total uncertainty is small. We find excellent agreement between experiment and the SM prediction for the mass difference: where we have used the input values given in Appendix B. The 10% uncertainty of the SM prediction is dominated by the hadronic and parametric CKM uncertainties which are of the same size. We also give results for the mass difference in the where the agreement is at the level of 1.1 sigma.
We determine the decay rate difference and the semileptonic decay asymmetry in the MS, PS [71], 1S [72] and kinetic [73] The different mass schemes are in good agreement with each other and we adopt the PS mass scheme as our central result. The SM value for the decay rate difference is in good agreement with the experimental average. The theory uncertainty is currently at the level of 30%. It is dominated by the matrix elements of the dimension seven operators, in particular the VSA estimate B R 2 = 1 ± 0.5 contributes ±0.016 ps −1 to the uncertainty. The second largest contribution is the scale variation. A detailed overview is given in Appendix B. To achieve a significant reduction of the combined uncertainties, a determination of the dimension seven matrix elements and a NNLO calculation of the perturbative matching are needed.
The experimental uncertainty for the semileptonic decay asymmetry is two orders of magnitude larger than the SM prediction, which makes this a clear null test for the SM [74]. The decay rate difference and the semileptonic decay asymmetry in the B d system have also not been measured yet. The current experimental averages and our predictions are The results obtained in different mass schemes are compatible and the relative uncertainties of the predictions are of the same magnitude as in the B s system.

∆B = 0 operators and ratios of B-meson lifetimes
The dominant contribution to lifetime differences between the mesons B q with q = u, d, s is due to spectator effects which first appear as dimension-six contributions in the HQE. The NLO Wilson coefficients have been computed in [75][76][77]. The dimension seven contributions are known at LO [29,78]. We define the set of operators in Section 5.1 and present the results for their Bag parameters in Section 5.2. The updated HQE results for the B-meson lifetime ratios are given in Section 5.3.

Operators and matrix elements
The following QCD operators enter at dimension six: On the HQET side they match ontõ Our basis of evanescent operators and the results of the matching computation can be found in Appendix A.2. We only consider the isospin-breaking combinations of operators and their analogues in HQET. This implies that the eye contractions displayed in Figure 5 cancel in the limit of exact isospin symmetry. The matrix elements are where Q = B − |Q|B − , the coefficients read Figure 5: Leading order eye contraction.
and B i = 1, i = 0 corresponds to the VSA approximation. Similarly we obtain for the HQET operators

Results for the spectral functions and bag parameters
For the ∆B = 0 operators we use the same conventions for the decomposition of the three-point correlator and the sum rule as for the ∆B = 2 operators above. We obtain for the double discontinuities of the non-factorizable contributions The leading condensate contributions have been determined in [14]. From their results we deduce that where the dots indicate factorizable contributions, α s corrections and contributions from condensates of dimension six and higher. To determine the condensate contributions to the HQET parameters we have used the traditional form of the sum rule, because the appearance of the δ-functions obviously prevents the application of a weight function analogous to (46). We find The associated errors were determined from an uncertainty of ±0.002 for missing higher-dimensional condensates, variations of the Borel parameters and the continuum cutoff and the uncertainty in the condensates α s π G 2 = (0.012 ± 0.006) GeV 4 , g sq σ µν G µν q = (−0.011 ± 0.002) GeV 5 .
(72) We note that our results for the contributions of the condensate corrections to the deviation of the Bag parameters from the VSA are much smaller than those of [14]. This is mostly due to the choice of the Borel parameter. We use t ∼ 1 GeV where the sum rule is stable against variations of the Borel parameter, while the Borel region of [14] translates to t = (0.35 − 0.5) GeV where the sum rule becomes unstable as can be seen in their plots. Our choice is also preferred by other modern sum rule analyses [12,13,80,82].
Following analysis strategy for the perturbative contributions described for the ∆B = 2 Bag parameters in Section 4.1, we find the HQET Bag parameters (73) where we have set a 1 = a 2 = 0. At the considered order there is no deviation from the VSA for the Bag parameters of the color singlet operators, as can be seen in (69) and (70), because the corresponding color factors vanish. The deviations for the color octet operators are in the range 0 -2 % for scales µ ρ between 1 and 2 GeV. In QCD we obtain  [14] and CY'98 [15], and the lattice values of UKQCD'98 [22] and Becirevic'01 [23]. The RG evolution and the perturbative matching cause larger deviations from the VSA which, however, do not exceed 11%. In Figure 6 we compare our results to previous ones from sum rules [14,15] and the lattice [22,23]. The results of [14,15,22] were obtained within HQET. For the comparison we match their results to QCD at tree level while expanding factors ofÃ i /A Q (m b (m b )) in 1/m b . As discussed in Section 4.1 this effectively includes 1/m b corrections in the VSA approximation. The B i are in good agreement, with the exception of the value for B 2 from [23], which differs from the other results and the VSA by a factor of about two. While the other sum rule results for the i agree reasonably well with ours, the lattice results for 1 show significantly smaller deviations from the VSA. The similarity between the sum rule results [14,15] and ours appears to be mostly coincidental. As discussed above, we find that the bulk of the deviation from the VSA in the i is due to the RG running and matching, while the latter was not considered in [14,15]. In their analyses, there is instead a sizeable deviation at the hadronic scale, originating from the condensate contributions. In comparison with [14] we find that this is due to the choice of very small values of the Borel parameter which lie outside of the stability region as discussed above. The assessment of the origin of the smallness of the lattice results [22,23] for the i is beyond the scope of this work. Many of the approximations made in [22,23], like quenching, have since been reappraised and a comparison with a state-of-the art lattice simulation is required.

Results for the lifetime ratios
Using our results (74) for the dimension-six Bag parameters and the VSA for the dimension-seven Bag parameters defined in [29], ρ i = 1 ± 1/12, σ i = 0 ± 1/6, we find excellent agreement with the experimental value and very good consistency between different mass schemes. The biggest contributions to the total uncertainty are still from the hadronic matrix elements, specifically from 1 with ±0.015 and σ 3 with ±0.013. In the future, they can be reduced with an independent determination of the dimension-six Bag parameters and a sum-rule determination of the dimensionseven Bag parameters. We also update the prediction for the lifetime ratio τ (B 0 s )/τ (B 0 ) in the MS scheme using Eq. (117) from [16]: where we have added an uncertainty estimate for the spectator effects at order 1/m 4 b which have not been considered in [16]. With respect to last year [6], the difference between the theory prediction and the experimental value for τ (B 0 s )/τ (B 0 ) is reduced from 2.5 σ to 1.1 σ.
6 Matrix elements for charm and the D + − D 0 lifetime ratio The HQET sum rule analysis can easily be adapted to the charm sector. It is common to quote the matrix elements for the charm sector at the scale 3 GeV instead of the charm-quark mass, see [30][31][32], and we adopt that convention for ease of comparison. Consequently we also use 3 GeV as the central matching scale. In the error analysis it is varied between 2 and 4 GeV. To account for the lower value of charm-quark mass we assume that the uncertainty due to power corrections is 0.03 instead of 0.01 for the bottom sector. Otherwise we use the same analysis strategy as in the bottom sector which is outlined in Section 4.1.

Matrix elements for D mixing
The latest lattice QCD study [32] for D mixing only gives results for the matrix elements and not for the Bag parameters. We do the same here and obtain, using the value of the D-meson decay constant from Appendix B, The relative uncertainties in the charm sector are consistently larger than those in the bottom sector because of larger perturbative corrections due to a larger value of α s at the smaller scales and larger power corrections. This effect is most pronounced for Q 2 , Q 4 and Q 5 where the relative uncertainty is larger by a factor of order two.
In the matrix elements we have an additional uncertainty from the value of the decay constant which is added in quadrature.  [30], ETM'15 [31] and FNAL/MILC'17 [32]. The values for the matrix elements of the ETM collaboration are extracted from Figure 16 of [32].
We compare our results to those from the lattice in Figure 7. There is a consistent hierarchy with decreasing values from the results of the FNAL/MILC collaboration [32], those of the ETM collaboration [30,31] and ours. The only exception is the value of Q 5 from [31] which lies below ours. If we use the lattice average [60] for the decay constant f lattice D = (211.9 ± 1.1) MeV in place of the experimental average f exp D = (203.7 ± 4.8) MeV [60], we find very good agreement between our results and those of ETM and the remaining differences with respect to the FNAL/MILC results are comfortably below two sigmas. We prefer the experimental average of the decay constant since it is in significantly better agreement with recent sum rule results [79][80][81][82]. On the other hand, using the lattice value yields a more meaningful comparison with the lattice results since the quantities we determine with the sum rule are the Bag parameters and the decay constant cancels out in the comparison if the same value is used on both sides. We therefore conclude that our sum rule results for the non-factorizable contributions to the Bag parameters are in good agreement with lattice simulations. An investigation of the differences in the numerical values of the decay constant is beyond the scope of this work. We update our result for the D-meson lifetime ratio from [29] using the dimension six Bag parameters (78) and the VSA ρ i = 1 ± 1/12, σ i = 0 ± 1/6 for the dimensionseven Bag parameters. We have converted the MS value of the charm-quark mass to the PS mass at µ f = 1 GeV and the 1S mass at four-loop accuracy using RunDec. The kinetic mass at the scale 1 GeV is determined with two-loop accuracy using an unpublished version of the QQbar Threshold code [83,84]. The central value for the scales µ 1 and µ 0 is fixed to 1.5 GeV for all mass schemes and varied between 1 and 3 GeV. We find which is in very good agreement. The various mass schemes are all consistent and we again take the PS result as our preferred value. The dominant sources of uncertainties are the Bag parameters 1 and σ 3 which both contribute ±0.5 to the error budget of the lifetime ratio. Both errors can be reduced in the future with a lattice determination of the dimensions-six matrix elements and a sum-rule determination of the dimension-seven Bag parameters, respectively. In the PS scheme, the radiative and power corrections are of the order +27% and −34%, respectively, which indicates good convergence behaviour. We therefore conclude that the HQE provides a good description of the lifetime ratio τ (D + )/τ (D 0 ).

Conclusions
We have determined the matrix elements of the dimension six ∆F = 0, 2 operators for the bottom and charm sector using HQET sum rules. Our findings for the ∆F = 2 matrix elements are in good agreement with recent lattice [3][4][5][30][31][32] and sum rule [11] results. Our ∆F = 0 results are the first state-of-the-art values for the matrix elements required for B and D meson lifetime ratios. The uncertainties in our analyses for the Bag parameters are similar to those of recent lattice determinations in the B sector and somewhat larger in the D sector. This suggests that the uncertainty of the ∆C = 0 matrix elements could be reduced by a lattice simulation. In most cases, the dominant errors in our approach stem from the matching of QCD to HQET operators, see Appendix B. These could be reduced substantially by performing the matching calculation at NNLO. Some first steps towards this goal have recently been taken in [70]. Consequently, in the future, sum rules will continue to be competitive with lattice simulations in the determination of four-quark operators. Our predictions for the mixing observables and lifetime ratios in the B sector are in good agreement with the experimental averages as summarized in Figures 8  and 9. In particular, the small tensions [5,6] that follow from using the FNAL/MILC results [5] for the matrix elements are not confirmed by our results. We note that the predictions based on matrix elements from sum rules and from lattice simulations are compatible and lead to overall uncertainties of the same size. Taking the naive average of the Bag parameters, the relative uncertainties of the mass and decay rate difference are, however, only reduced by about 9% and 6%, respectively, because other sources of uncertainties, like e.g. the matrix elements of dimension-seven operators, are dominant. . We also show the results obtained with the lattice results of [5] for f 2 Bq B Q i and the matrix element R 0 and the values given in Appendix B for the other input parameters. The PS mass scheme for the bottom quarks has been used in both cases.
We find that the experimental value for the lifetime ratio τ (D + )/τ (D 0 ) can be reproduced within the HQE. This is a strong indication that the HQE does not break down in the charm sector. However, due to sizeable hadronic uncertainties, we cannot exclude large duality violations at the level of 20-30% yet. On the other hand, the D-mixing observables are very sensitive to duality violations and might offer a handle on a better quantitative understanding of these effects [25].
Our comprehensive study demonstrates that the HQET sum rules for hadronic four-quark matrix elements provide a competitive alternative to lattice simulations. Due to completely different systematics they facilitate powerful independent checks of lattice results. Sum rules can also be applied to obtain the matrix elements of the subleading dimension-seven operators, which have never been determined using lattice simulations. This is crucial to achieve a substantial reduction of the current theoretical uncertainties.

A.1 ∆B = 2 operators
Our choice of basis for the evanescent operators is given by for QCD and for HQET. It is straightforward to verify that the evanescent operators vanish in four dimensions by using the Fierz identities and the relation γ µ γ ν γ ρ = g µν γ ρ + g νρ γ µ − g µρ γ ν − i µνρλ γ λ γ 5 .
A useful strategy to simplify expressions with two Dirac matrices is to use projection identities, e.g.h and then reduce the number of Dirac matrices with Eq. (83).
In the decomposition (8) the LO QCD ADM is In HQET we find Our result (85) with N c = 3 differs from the results of [85,86] because we have only used the replacements implied by the basis of evanescent operators (80) to simplify products of Dirac matrices. We can reproduce their result by applying 4-dimensional Fierz identities that relate Q 1 , Q 2 and Q 3 . The upper left 2 × 2 submatrix of (87) agrees with [34].

A.2 ∆B = 0 operators
We define the basis of evanescent operators in QCD following [75]: In HQET we again introduce parameters a 1,2 to keep track of the scheme dependencẽ The isospin breaking combinations of the evanescent operators are defined in analogy to (64). The LO ADM in QCD takes the form The HQET result is given bỹ (94) Our result (91) is in agreement with [87,88] and (93) reproduces the result of [89]. 4 The results (92) and (94) Table 2: Individual errors for the Bag parameters of the ∆B = 2 matrix elements. 5 We take the mean of f B + and f B 0 . 6 We use the 'experimental' value instead of the lattice average, since the former is in significantly better agreement with sum rule results [79][80][81][82]       ±0.00 Table 9: Individual errors for the ratio τ (D + )/τ (D 0 ) in the PS mass scheme.